Towards a Variational Approach to Regularized Tomographic Reconstruction

The classical Kaczmarz’s method, which is the basis for many Algebraic Reconstruction Techniques (ART), is very popular in the field of image reconstruction. However, this algorithm gives only satisfactory results for consistent data. For inconsistent data, which is the case in practice, the algorithm has some problems with respect to convergence. In this master’s thesis, we formulated the reconstruction problem in the framework of variational calculus. Additional to the reconstruction part, we introduced a regularization term which enforces the overall smoothness in the image. We examined five different regularization terms; first of all, the linear regularizer which we call Quadratic function, then two convex regularizers which we call Hypersurface minimal and Green function, and finally, two non-convex terms which we denominate Lorentzian and Tukey function. If the right parameters are chosen the four non-linear regularizers intensify smoothness within homogenous areas while edges are preserved. Out of the variational approach, we developed three extensions of the Kaczmarz’s algorithm; first of all, the Regularized Kaczmarz Extended with Relaxation Parameters algorithm (RKERP), secondly, a simplified version of it (SRKERP), and finally, the new Regularized Algebraic Reconstruction Technique algorithm (RART). The algorithms were examined for the reconstruction of three-dimensional (3-D) medical image data out of only a few two-dimensional (2-D) X-ray images. Our experiments showed that only the SRKERP algorithm yields satisfactory results.